Numbers

Numeral

A numeral is a symbol, or group of symbols, that we use to write down a number. For example, the following are all numerals:

The Arabic numerals we use ten digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Roman numerals, like VI (for 6) or XCVIII (for 98)
Binary digits, 0 and 1, used in computers

Place Value

Place value refers to the value of the position of a digit in a number. The position of a digit determines its place value, and the place values are powers of 10. The place values are:

Units (or Ones): 1
Tens: 10
Hundreds: 100
Thousands: 1,000
Ten Thousands: 10,000
Hundred Thousands: 100,000
Millions: 1,000,000
and so on…

Face Value

The face value of a digit in a number is the value of the digit itself, irrespective of its position in the number.

Let’s take a number, for example, 3456:

The face value of 3 is 3.
The face value of 4 is 4.
The face value of 5 is 5.
The face value of 6 is 6.

The place value of each digit in this number is:

6 is in the units place, so its place value is 6
5 is in the tens place, so its place value is 50
4 is in the hundreds place, so its place value is 400
3 is in the thousands place, so its place value is 3000

Types of Numbers

Sure, here’s a straightforward explanation of the types of numbers in point form, with relevant words highlighted:

Natural Numbers:

Counting numbers starting from 1.
Example: 1, 2, 3, 4, 5, …

Whole Numbers:

Natural numbers along with zero.
Example: 0, 1, 2, 3, 4, 5, …

Integers:

Whole numbers and their negative counterparts.
Example: … -3, -2, -1, 0, 1, 2, 3, …

Rational Numbers:

Numbers that can be expressed as a fraction of two integers.
Example: 1/2, 3/4, -5/3, 7, …

Irrational Numbers:

Numbers that cannot be expressed as a fraction and have non-repeating and non-terminating decimals.
Example: √2, π, e, …

Real Numbers:

All rational and irrational numbers combined.
Example: -5, 0, 1.5, √2, π, …

Imaginary Numbers:

Numbers that involve the square root of a negative number.
Example: √(-1), i, 2i, …

Complex Numbers:

Numbers that have both real and imaginary parts.
Example: 3 + 2i, 4 – 5i, …

Tests of Divisibility

Divisibility by 2

Test: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
Example:
- 14 (ends with 4, so divisible by 2)
- 37 (ends with 7, so not divisible by 2)

Divisibility by 3

Test: A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
- 123 (1 + 2 + 3 = 6, divisible by 3)
- 128 (1 + 2 + 8 = 11, not divisible by 3)

Divisibility by 4

Test: A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
Example:
- 148 (48 is divisible by 4, so 148 is also divisible by 4)
- 135 (35 is not divisible by 4, so 135 is not divisible by 4)

Divisibility by 5

Test: A number is divisible by 5 if its last digit is either 0 or 5.
Example:
- 105 (ends with 5, so divisible by 5)
- 128 (does not end with 0 or 5, so not divisible by 5)

Divisibility by 6

Test: A number is divisible by 6 if it is divisible by both 2 and 3.
Example:
- 24 (divisible by 2 and 3, so divisible by 6)
- 21 (divisible by 3 but not by 2, so not divisible by 6)

Divisibility by 9

Test: A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
- 162 (1 + 6 + 2 = 9, divisible by 9)
- 148 (1 + 4 + 8 = 13, not divisible by 9)

Divisibility by 10

Test: A number is divisible by 10 if it ends with 0.
Example:
- 120 (ends with 0, so divisible by 10)
- 128 (does not end with 0, so not divisible by 10)

Divisibility by 11

Test: A number is divisible by 11 if the difference between the sum of its odd-positioned digits and the sum of its even-positioned digits is either 0 or a multiple of 11.
Example:
- 143 (1 + 4 = 5, 3 = 3, difference is 2, not divisible by 11)
- 132 (1 + 2 = 3, 3 = 3, difference is 0, divisible by 11)

MCQ’s

1. Which of the following is a prime number?
A) 16
B) 21
C) 29
D) 42
Answer: C) 29

2. What is the next number in the sequence: 2, 4, 8, 16, ?
A) 24
B) 18
C) 32
D) 30
Answer: C) 32

3. Which of the following is a composite number?
A) 7
B) 9
C) 11
D) 13
Answer: B) 9

4. What is the square root of 81?
A) 7
B) 8
C) 9
D) 10
Answer: C) 9

5. Which of these numbers is a multiple of 5?
A) 32
B) 25
C) 49
D) 36
Answer: B) 25

6. What is the sum of the first 5 prime numbers?
A) 15
B) 18
C) 28
D) 30
Answer: C) 28

7. Which of the following numbers is an odd number?
A) 42
B) 50
C) 63
D) 72
Answer: C) 63

8. What is the next number in the sequence: 1, 4, 9, 16, ?
A) 20
B) 24
C) 25
D) 36
Answer: C) 25

9. Which number comes between 5 and 7?
A) 5.5
B) 6
C) 6.5
D) 7.5
Answer: B) 6

10. What is 100 divided by 10?
A) 10
B) 20
C) 50
D) 90
Answer: A) 10

11. Which number is not a factor of 12?
A) 2
B) 3
C) 4
D) 5
Answer: D) 5

12. What is 5 times 6?
A) 25
B) 30
C) 35
D) 40
Answer: B) 30

13. What is the smallest 2-digit prime number?
A) 10
B) 11
C) 12
D) 13
Answer: B) 11

14. What is the next number in the sequence: 3, 6, 12, 24, ?
A) 28
B) 36
C) 48
D) 50
Answer: C) 48

15. Which number is a perfect square?
A) 45
B) 49
C) 50
D) 52
Answer: B) 49

16. What is 7 multiplied by 0?
A) 0
B) 1
C) 7
D) 14
Answer: A) 0

17. Which of the following numbers is the largest?
A) 0
B) -1
C) 1
D) -2
Answer: C) 1

18. What is the value of 10 squared?
A) 100
B) 110
C) 120
D) 90
Answer: A) 100

19. Which of these numbers is a prime number?
A) 22
B) 25
C) 29
D) 30
Answer: C) 29

20. What is the product of 8 and 7?
A) 48
B) 54
C) 56
D) 64
Answer: C) 56